Forecasting from others’ experience: Bayesian estimation of the generalized Bass model

International Journal of Forecasting xxx (xxxx) xxx

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Forecasting from others’ experience: Bayesian estimation of the generalized Bass model Andrés Ramírez-Hassan a , Santiago Montoya-Blandón b,a , a b

∗

Department of Economics, Universidad EAFIT, Medellín, Carrera 49 - 7 S 50, Colombia Department of Economics, Emory University, Rich Building, 1602 Fishburne Dr., Atlanta, GA 30322-2240, USA

article

info

Keywords: Bayesian estimation Diffusion Forecast by analogy Generalized Bass model Pre-launch forecast

a b s t r a c t We propose a Bayesian estimation procedure for the generalized Bass model that is used in product diffusion models. Our method forecasts product sales early based on previous similar markets; that is, we obtain pre-launch forecasts by analogy. We compare our forecasting proposal to traditional estimation approaches, and alternative new product diffusion specifications. We perform several simulation exercises, and use our method to forecast the sales of room air conditioners, BlackBerry handheld devices, and compressed natural gas. The results show that our Bayesian proposal provides better predictive performances than competing alternatives when little or no historical data are available, which is when sales projections are the most useful. © 2019 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

1. Introduction Diffusion models have been found to be remarkably important in several disciplines, and have been crucial in the study of the introduction of new products (Sultan, Farley, & Lehmann, 1990). A considerable amount of academic research has addressed the subject, covering a wide range of different products, marketing strategies, forecasting procedures, and interactions among market forces (excellent reviews are provided by Bridges, Coughlan, & Kalish, 1991; Mahajan, Muller, & Bass, 1990; Meade & Islam, 2006; Parker, 1994; Rao & Kishore, 2010). Of the many models that have been used for studying product diffusion, the Bass model (BM) (Bass, 1969) stands out as the one that has been studied most extensively (Peres, Muller, & Mahajan, 2010). This model is particularly appealing due to its ability to replicate and forecast sales data for a broad array of products through simple mathematical and statistical concepts. It was also the first model ∗ Corresponding author at: Department of Economics, Universidad EAFIT, Medellín, Carrera 49 - 7 S 50, Colombia. E-mail addresses: [email protected] (A. Ramírez-Hassan), [email protected] (S. Montoya-Blandón).

to include consumers’ behavioral assumptions in the diffusion framework explicitly using the ideas of Rogers (1962). The BM relies on some concepts from survival analysis, specifically the hazard rate. This quantity is defined as the rate of adopting a product or service, conditional on not having adopted it previously. The hazard rate of adoption is assumed to be related linearly to the fraction of total buyers from the potential market of a product (Hong, Koo, & Kim, 2016). The intuition is that a certain segment of the population, known as the ‘‘innovators’’, will adopt the product regardless of who has bought it before, and then convey the information through interpersonal communication (word-of-mouth) to the ‘‘imitators’’, who adopt the product later. The behavioral assumptions that underlie the model are contained in three key parameters: the innovation coefficient (p), the imitation coefficient (q), and the short-run potential market (m). The BM is regarded as one of the most important ideas in marketing research (Bass, 2004; Hopp, 2004). However, it has not been deemed flawless; for instance, it fails to take into account left-hand truncation (Jiang, Bass, & Bass, 2006; Kim & Hong, 2015) and assumes constant market

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Please cite this article as: A. Ramírez-Hassan and S. Montoya-Blandón, Forecasting from others’ experience: Bayesian estimation of the generalized Bass model. International Journal of Forecasting (2019), https://doi.org/10.1016/j.ijforecast.2019.05.016.

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potential (Guseo, Dalla, Furlan, Guidolin, & Mortarino, 2017; Guseo & Guidolin, 2009). One of its major shortcomings is that, initially, it does not allow the inclusion of variables such as price, advertisement, or promotion, which play an important role in determining the eventual adoption by consumers (Schmittlein & Mahajan, 1982). While many authors have introduced modifications to the Bass model or their own models that take this issue into account, one of the most elegant solutions, referred to as the generalized Bass model (GBM), was introduced by Bass, Krishnan, and Jain (1994). Their model extends the original Bass model to include market effort, which comprises a linear combination of the growth rate of the price, advertisement, and possibly other variables. They showed that the GBM reduces to the standard Bass model if these variables are not significant in explaining the adoption rate or if their growth rate remains constant over time. Furthermore, we should take into account the fact that the marketing effort at time t has an effect not only on the sales at t, but also on the sales in future periods (Krishnan & Jain, 2006). This goodwill effect is different from the word-of-mouth effect in the Bass model. The GBM has been found to perform well at forecasting sales curves for various products (Chandrasekaran & Tellis, 2007; Dalla Valle & Furlan, 2011), and therefore will be the focus of our analysis. Given sufficient data (i.e., data up to and including the sales peak for a given product), traditional estimation approaches can be used to estimate the BM and the GBM (Mahajan, Mason, & Srinivasan, 1986). Bass (1969) introduced a model that could be estimated by ordinary least squares (OLS) after transforming the solution of the model from continuous to discrete time. Schmittlein and Mahajan (1982) proposed a maximum likelihood (ML) approach based on the unconditional probabilities of adoption that corrected for time aggregation bias and allowed the computation of standard errors for the parameters. Srinivasan and Mason (1986) introduced a nonlinear least squares (NLS) method that estimates better standard errors for the parameters and is usually the preferred estimation approach in the diffusion literature (see Mahajan et al., 1986, for a comparison of the three methods). Recently, Hong et al. (2016) proposed a hybrid approach of NLS and OLS (HON). Using both simulations and real data sets, they showed that their method is accurate and stable. Historical sales information and forecasts are extremely valuable to marketers (Rossi, McCulloch, & Allenby, 1996). From the point of view of a manager who is planning to introduce an innovation into the market, demand forecasts are fundamental for planning the production, distribution, and marketing efforts for a new product. Thus, obtaining reliable and accurate forecasts prior to (pre-launch) or during the initial periods of a launch is a crucial element of product planning. However, the main concern is that one needs sufficient data in order to estimate the diffusion models for forecasting the sales curve, and by the time enough data have been collected, the product has reached maturity and the forecasts are rendered useless (Goodwin, Dyussekeneva, & Meeran, 2013; Guseo et al., 2017; Kim, Hong, & Koo, 2013; Lenk & Rao,

1990; Mahajan et al., 1990). For this reason, researchers have proposed various pre-launch forecasts based on subjective approaches driven by expert surveys (Kim et al., 2013), judgments by potential customers (Bass, Gordon, Ferguson, & Githens, 2001), or guessing by analogy in either a static (Goodwin et al., 2013; Wright & Stern, 2015) or a dynamic (Guseo et al., 2017) version. However, these approaches do not take into account parameter uncertainty (standard errors) explicitly, and do not allow consistent parameter updating following rules of probability. Therefore, other researchers have turned to Bayesian techniques, since it fits the problem structure naturally: it systematically incorporates prior information from experts or similar products to make predictions, and then updates these predictions as data become available (Kulkarni, Kannan, & Moe, 2012; Lee, Boatwright, & Kamakura, 2003; Lenk & Rao, 1990; Lilien, Rao, & Kalish, 1981; Mahajan et al., 1990; Putsis Jr. & Srinivasan, 2000; Van Everdingen, Aghina, & Fok, 2005). The main objective of the present paper is to propose a Bayesian method based on the GBM for forecasting the sales of new products. The method allows flexibility by incorporating marketing effort, estimating the GBM before sales reach their peak, and producing forecasts when no data are available (pre-launch). We used guessing by analogy (Goodwin et al., 2013; Guseo et al., 2017; Wright & Stern, 2015), such that the prior distribution is based on reference markets. The adaptation of the posterior distribution as data become available is rapid: we can obtain accurate forecasts before the sales peak is reached. Our Bayesian approach allows us to obtain predictive intervals that take into account parameter uncertainty without extra analytic or computational effort. The literature regarding predictive intervals in diffusion models is scarce (Goodwin, Meeran, & Dyussekeneva, 2014), with exceptions being the studies by Meade (1985), Meade and Islam (1995) and Migon and Gamerman (1993). The remainder of this paper is structured as follows: Section 2 introduces the method and some of the innovations with which we propose to estimate the model using a Bayesian approach. Section 3 presents some simulation results that compare our approach to others that are used in the literature. Section 4 presents the results of using the method with the GBM in three applications: room air conditioners, BlackBerry handheld devices and compressed natural gas. Lastly, Section 5 presents our concluding remarks.

2. The method The usual representation of the Bass model is f (t) 1 − F (t)

=p+

q m

Y (t),

(1)

where the parameters p, q, and m are interpreted as stated previously. Here, F (t) and f (t) are the cumulative

Please cite this article as: A. Ramírez-Hassan and S. Montoya-Blandón, Forecasting from others’ experience: Bayesian estimation of the generalized Bass model. International Journal of Forecasting (2019), https://doi.org/10.1016/j.ijforecast.2019.05.016.

A. Ramírez-Hassan and S. Montoya-Blandón / International Journal of Forecasting xxx (xxxx) xxx

and non-cumulative proportions of adopters at time t, and Y (t) is the total number of adopters up to but not including time t. We derive the generalized Bass model by incorporating market effort into the BM framework. This addition is introduced as a multiplicative term on the right hand side of Eq. (1): f (t) 1 − F (t)

[ ] q = p + Y (t) x(t),

(2)

m

where x(t) is the market effort, usually assumed (Bass et al., 1994; Simon, 1982) to be x(t) = 1+β1

P(t) − P(t − 1) P(t − 1)

+β2

max {0, A(t) − A(t − 1)} A(t − 1)

,

(3) where P(t) and A(t) are the price and the advertising, respectively. These variables enter the market effort equation as percentage increases. The inclusion of other variables is then trivial. We propose a Bayesian method for estimating and forecasting the GBM, which fits naturally into the innovation diffusion context. As with all Bayesian models, we will need to supply a likelihood function and prior distributions, to be combined using Bayes’ rule (Rossi & Allenby, 2003). For the likelihood function, we will use the sales function proposed by Srinivasan and Mason (1986). By using the appropriate time aggregation of Schmittlein and Mahajan (1982), the sales S(t) between times t and t − 1 are defined as S(t) = m[F (t) − F (t − 1)] + ut ,

(4)

where ut is an additive error term with variance σ 2 , and F (t) is the solution to the GBM, given by (see the appendix) F (t) =

1 − exp{−X (t)(p + q)} 1 + (q/p) exp{−X (t)(p + q)}

,

(5)

where X (t) is the cumulative market effort, found by transforming Eq. (3) into continuous time and integrating from 0 to t: X (t) = t + β1 ln

[

P(t) P(0)

]

[ + β2 ln

ˆ A(t) A(0)

] ,

(6)

ˆ is the last value for which there was a positive where A(t) change in advertising. In more general terms, Eq. (6) can be written as X (t) = t + β′ Z˙ (t),

(7)

where β is a k-dimensional vector of parameters, Z (t) is a k-vector of covariates at time t and Z˙ (t) = [ln(Z1 (t)/ Z1 (0)), . . . , ln(Zk (t)/Zk (0))]′ . We assume that ut is normally distributed, and so the likelihood function is also normal. With T intervals of data and writing S, F , and Z for the collection of sales, cumulative adoption, and marketing effort variables, respectively, the likelihood

3

function can be expressed as f (S |p, q, m, β, σ 2 , Z ) =

(

1

)T /2 ×

2πσ 2

{

1

[S − m∆F (p, q, β)]′ } × [S − m∆F (p, q, β)] . exp −

2σ 2

(8) Writing θ = [p, q, m, β ] , we assume that π (θ ) is a multivariate t-distribution with location prior hyperparameters θ 0 , prior scale matrix Σ0 , and ν0 = 3 degrees of freedom. This choice of the Bass model parameters, which includes only a few degrees of freedom, implies fatter tails, and therefore robustness of the outcomes (Berger, 1985).1 For the variance parameter σ 2 , we decide against using the standard, inverse gamma IG (ϵ, ϵ ) distribution with ϵ → 0 (Kelsall & Wakefield, 1999; Lunn, Spiegelhalter, Thomas, & Best, 2009). As was shown by Gelman (2006), this distribution does not actually possess a proper limiting posterior distribution, and should be avoided in empirical work. Instead, we will use the Scaled-Beta2(a, b, κ ) distribution that was proposed by Fúquene, Pérez, and Pericchi (2014) and introduced to the econometric literature by Ramírez Hassan and Pericchi (2018).2 This distribution can be derived from a mixture of gamma distributions and is easy to simulate from the scaled odds of a beta distribution (Fúquene et al., 2014, see the appendix). We follow Fúquene et al. (2014) and set the prior hyperparameters a0 = b0 = 1 and set κ0 to a large number. As was pointed out by Pérez, Pericchi, and Ramírez (2017), when a = b, κ becomes the median of the variance prior, so that increasing κ gives an arbitrarily large a priori variance in the model. Preliminary simulation exercises using the inverse gamma distribution showed poor performances relative to the Scaled-Beta2 distribution, which further motivated our selection (see Table 4 in the text and Table B.3 in the appendix). The prior distribution for the GBM is given by ′ ′

π (θ, σ 2 ) = π (θ )π (σ 2 ) = t(θ 0 , Σ0 , ν0 )SBeta2(a0 , b0 , κ0 ).

(9)

For estimation purposes, we set the prior parameters p0 , q0 , m0 , and β0 , as well as Σ0 , from previous, similar markets (Lee et al., 2003; Lenk & Rao, 1990; Lilien et al., 1981; Parker, 1994). Even in the event that no data are available, we can draw values for the parameters from the priors and use them to predict future sales.

1 Note that we do not impose prior theoretic restrictions on θ , such as 0 < p < 1. However, imposing such restrictions can be done easily in our framework (see Section 4.3). 2 A generalized version of this distribution has become known in the statistics literature as the generalized beta prime distribution (Dubey, 1970).

Please cite this article as: A. Ramírez-Hassan and S. Montoya-Blandón, Forecasting from others’ experience: Bayesian estimation of the generalized Bass model. International Journal of Forecasting (2019), https://doi.org/10.1016/j.ijforecast.2019.05.016.

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Table 1 Values for simulation exercises. Population Market 1 Market 2 Market 3

p

q

m

M

β1

β2 β3

σ2 κ

0.010 0.010 0.020 0.025

0.40 0.35 0.40 0.50

20,000 20,000 30,000 50,000

50,000 50,000 50,000 50,000

−1.5 −1.3 −1.0 −0.2

1.0 1.0 1.5 3.5

1 1 1 1

−2.5 −1.0 −4.0 −2.5

2000 2000 2000 2000

Notes: Population parameters in simulation exercises.

Combining the likelihood function and the prior distribution using Bayes’ rule, we obtain the posterior distribution for the GBM parameters

π (θ, σ 2 |S , Z ) ∝

(

1

)T /2

2π σ 2

{ 1 × exp − 2 [S − m∆F (p, q, β)]′ 2σ } × [S − m∆F (p, q, β)] ×

Γ (ν0 /2 + 2) |Σ0 |1/2 (ν0 π )2 Γ (ν0 /2)

[ ]−ν0 /2−2 1 × ν0 + (θ − θ 0 )′ Σ− 0 (θ − θ 0 ) ×

Γ (a0 + b0 ) 1 (σ 2 /κ0 )a0 −1 . 2 Γ (a0 )Γ (b0 ) κ0 (σ /κ0 + 1)a0 +b0 (10)

3. Simulation exercises The focus of this section is on presenting exercises which compare the goodness-of-fit of several models on a simulated data set. Defining population values for p, q, m and β allows us to find F (t), X (t) and S(t). We compare our proposed method to more classical alternatives, which will help us to discern the gains in forecasting power of our Bayesian estimation approach. Analysis and comparisons to these estimation methods, as well as other modern specifications, is delayed until the next section, in the context of the applications we consider, where it will be more useful. We assume that there are three covariates, and that these are controlled by the manager, meaning that the complete time path of the covariates is known at any time. The variables were generated from normal distributions with means of 4000, 1500 and 3000, and standard deviations of 100, 25 and 50, respectively. We then generated a few random previous markets and obtained the prior hyperparameters from them. The location vector and scale matrix were calibrated using the means and variances, respectively, of each parameter across previous markets. The reference values can be observed in Table 1. The population values for the simulation generate sales data with takeoff at t = 5, peak at t = 10, and saturation point at t = 13. We included the long-run market M, which is needed only for maximum likelihood estimation. With this level of M, the short-run potential market is 40% of the long-run analog. We began by

estimating the model at t = 0, 5, 8, 9, 10, 15, 20, 25 using our Bayesian method, the NLS technique of Srinivasan and Mason (1986), and the ML approach of Schmittlein and Mahajan (1982). We estimated the Bayesian model using a Metropolis–Hastings sampling algorithm with three parallel chains, each consisting of 50,000 iterations, a burn-in period of 25,000, and samples extracted every five iterations so as to avoid autocorrelation in the chains (Robert & Casella, 2004). All of the chains were checked for convergence using standard tests and were found to fulfill all convergence criteria (Brooks & Gelman, 1998; Geweke, 1992; Heidelberger & Welch, 1983; Raftery & Lewis, 1992). A second Bayesian alternative with non-informative priors, which have θ 0 = 0 and Σ0 = 1, 000I , was computed under the same conditions. We estimated the NLS model using three different algorithms: unconstrained optimization based on the Nelder–Mead algorithm (Nelder & Mead, 1965), the PORT subroutine library for constrained optimization (Fox, Hall, & Schryer, 1978), and a genetic algorithm that does not require initial values (Sekhon & Mebane, 1998).3 All of the procedures returned similar values, and therefore we present only those obtained using the PORT algorithm. Finally, we estimated the ML method using the Nelder–Mead algorithm explicitly including the parameter restrictions in the objective function. The results are presented in Table B.1 of the appendix. The point estimates and standard errors of the Bayesian model are the means and standard deviations, respectively, of the final chains. For the frequentist approaches, the usual coefficients and standard errors were computed. Furthermore, we calculated the peak, takeoff and saturation points of the fitted sales at each information horizon, as these are of great managerial importance. While the characterization of these quantities in the context of the GBM (Bass et al., 1994) is simple, explicit values in terms of the parameters are not easily available. We use the more modern and operational definitions of Hong et al. (2016) rather than the heuristic introduced by Golder and Tellis (1997) and applied by Sood, James, and Tellis (2009), where the sales peak is computed as the maximum of the fitted noncumulative sales, while the takeoff and saturation points are computed as the inflection points of the sales curve.4 Curiously, in this simulation exercise, the frequentist approach yields good estimation results at t = 8, a period before the sales peak is reached. However, the impossibility of estimating the model in previous periods gives the Bayesian framework a clear advantage in terms of actual product planning. Maximum likelihood estimates are available with small amounts of information, but standard errors are not available for all of the parameters when t = 5. It appears that the maximum likelihood method is particularly good for estimating the standard Bass model parameters, but the coefficients for 3 All computations were performed using the R open-source programming language (R Core Team, 2015). Specific packages include those of Plummer, Best, Cowles, and Vines (2006), Genz et al. (2014), Plummer (2015), Su and Yajima (2015), and Christopoulos (2017). 4 We calculate these using the methods proposed by Christopoulos (2016).

Please cite this article as: A. Ramírez-Hassan and S. Montoya-Blandón, Forecasting from others’ experience: Bayesian estimation of the generalized Bass model. International Journal of Forecasting (2019), https://doi.org/10.1016/j.ijforecast.2019.05.016.

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Table 2 Comparison of one-step-ahead mean squared errors (MSE) and mean absolute errors (MAE) between Bayesian and frequentist estimation with different amounts of information. t=0

t=5

t=8

t=9

t = 10

t = 15

t = 20

t = 24

0.10 0.32

0.20 0.48

0.90 0.91

2.72 1.65

0.01 0.12

Noncumulative sales Bayesian estimation: informative prior MSE MAE

2.65E+05 515.20

5.44 2.33

0.02 0.15

Bayesian estimation: non-informative prior MSE MAE

6.12E+04 247.57

– –

813.97 28.53

1.52 1.23

15.36 3.92

1.29 1.14

2.80 1.67

0.01 0.12

– –

36.45 6.04

5.28 2.30

14.85 3.85

1.45 1.20

2.77 1.66

0.01 0.12

949.32 30.81

79.33 8.91

11.19 3.35

7677.53 87.62

4.99 2.23

4.59 2.14

0.02 0.13

0.15 0.38

0.02 0.13

0.49 0.70

0.63 0.79

0.35 0.59

NLS estimation MSE MAE

– –

ML estimation MSE MAE

– –

Cumulative sales Bayesian estimation: informative prior MSE MAE

2.65E+05 515.20

12.5 3.53

0.12 0.35

Bayesian estimation: non-informative prior MSE MAE

6.12E+04 247.57

– –

7.33 2.71

11.64 3.41

12.70 3.56

0.03 0.17

1.13 1.06

0.35 0.59

– –

35.72 5.98

7.00 2.64

13.58 3.68

0.25 0.50

1.19 1.09

0.40 0.64

1601.67 40.02

82.49 9.08

2479.31 49.79

9776.02 98.87

13 528.77 116.31

5859.80 76.55

160.81 12.68

NLS estimation MSE MAE

– –

ML estimation MSE MAE

– –

Notes: The models are fitted on each subsample up to t. Informative priors use information from previous markets, while non-informative priors are centered at zero with a high variance. NLS and MLE include appropriate parameter restrictions. Minimum values are shown in bold. ‘‘–’’ represents unavailable estimation results.

the short-run potential market and marginal effects are much more volatile and present higher standard errors than the other methods. Similar trends are maintained when focusing on the other sales measures of peak, takeoff and saturation. While all methods seem to perform well after the peak of sales has been reached, the informative Bayesian alternative provides a good approximation to all measures at t = 5. Using these estimation results, we compare the goodness-of-fit of the estimated models using one-stepahead forecasts of sales, both yearly and cumulative. These results can be observed in Table 2, which presents the mean squared errors (MSE) and mean absolute errors (MAE) for the one-step-ahead forecasts. Blank spaces arise from the impossibility of computing estimates of the parameters for particular horizons. In general, the forecasting error remains small when more information is incorporated, as one would expect. However, this is not the case for maximum likelihood estimation, which, despite working for parameter estimation previously, produces poor one-step-ahead forecasts relative to the other methods, even with full information. Notice how the forecasting error is reduced by approximately 100% each time when going from t = 0 to t = 5, and from t = 5 to t = 8. To summarize, informative Bayesian estimation exhibits the lowest errors, except at t = 0. In addition,

informative priors help to achieve convergence when little information is available; non-informative Bayesian estimation does not achieve convergence at t = 5. As a second exercise, we then explored what would happen if the potential short-run market, m, was known. As was noted by Parker (1994), Trajtenberg and Yitzhaki (1989) and Van den Bulte and Lilien (1997), the estimation of the potential short-run market, in terms of either units or a percentage of a fixed long-run market, is what introduces the most noise into the estimation and forecasting procedure. Parker (1994), in particular, notes that this parameter should not be estimated using early data on sales, and that there are several methods that work well in practice for approximating this value, such as consumer intentions surveys, demographic data or management judgment (Bass et al., 2001). Eliminating m from θ and fixing it at 20,000 results in a vast improvement in both estimation and forecasting, as can be observed in Table B.2 and Figure B.3 in the appendix. Not having to estimate m makes both estimation procedures more accurate and computationally faster. Bayesian estimation still has the advantage of providing sensible estimates when historical information is lacking. However, we can observe that NLS is now able to provide point estimates for the parameters at t = 5, albeit without standard errors. The ML estimates for p and q remain quite stable,

Please cite this article as: A. Ramírez-Hassan and S. Montoya-Blandón, Forecasting from others’ experience: Bayesian estimation of the generalized Bass model. International Journal of Forecasting (2019), https://doi.org/10.1016/j.ijforecast.2019.05.016.

6

A. Ramírez-Hassan and S. Montoya-Blandón / International Journal of Forecasting xxx (xxxx) xxx Table 3 Comparison of one-step-ahead mean squared errors (MSE) and mean absolute errors (MAE) between Bayesian and frequentist estimation with different amounts of information without estimating the short-run market. t=0

t=5

t=8

t=9

t = 10

t = 15

t = 20

t = 24

0.10 0.32

0.20 0.48

0.90 0.91

2.72 1.65

0.01 0.12

Noncumulative sales Bayesian estimation: informative prior MSE MAE

2.65E+05 515.20

5.44E+00 2.33

0.02 0.15

Bayesian estimation: non-informative prior MSE MAE

1.87E+08 1.36E+04

– –

1.76 1.33

5.09 2.26

8.90 2.98

0.49 0.70

2.53 1.59

0.01 0.10

– –

36.45 6.04

5.28 2.30

14.85 3.85

1.45 1.20

2.77 1.66

0.01 0.12

1045.68 32.34

41,479.74 203.67

11,339.71 106.49

562.17 23.71

1522.46 39.02

25.08 5.01

0.98 0.99

0.15 0.38

0.02 0.13

0.49 0.70

0.63 0.79

0.35 0.59

NLS estimation MSE MAE

– –

ML estimation MSE MAE

– –

Cumulative sales Bayesian estimation: informative prior MSE MAE

2.65E+05 515.20

12.5 3.53

0.12 0.35

Bayesian estimation: non-informative prior MSE MAE

1.87E+08 1.36E+04

– –

0.19 0.44

13.54 3.68

1.05 1.02

24.37 4.94

17.58 4.19

18.44 4.29

23.46 4.84

1.05 1.03

7.23 2.69

8.77 2.96

0.54 0.73

2.52 1.59

0.01 0.10

1130.78 33.63

39,133.00 197.82

5449.01 73.82

628.39 25.07

34.41 5.87

13.65 3.70

14.63 3.82

NLS estimation MSE MAE

– –

ML estimation MSE MAE

– –

Notes: The models are fitted on each subsample up to t. Informative priors use information from previous markets while non-informative priors are centered at zero with a high variance. NLS and MLE include appropriate parameter restrictions. Minimum values are shown in bold. ‘‘–’’ represents unavailable estimation results.

but the marginal effects coefficients vary wildly when additional information is included. Table 3 presents the MSE and MAE values for this exercise. In general, the trends from the previous exercises are maintained. Figure B.3 shows how a similar m shapes the cumulative adoption curve. The forecasting accuracy is improved substantially by having the endpoints of all forecast curves being the same. We compare the results obtained in the previous exercises using a Scaled-Beta2 distribution with those generated by assuming the usual Inverse Gamma prior. Table 4 presents the mean squared and mean absolute errors associated with one-step-ahead predictions from each model. While the downward trend of the errors as more information is included in the analysis remains similar, it appears that the Scaled-Beta2 model is better in terms of forecasting. In most scenarios, one-step-ahead forecasts are improved slightly relative to this base model, and this difference is especially noticeable for most time periods before or exactly at t = 10. Lastly, Figs. 1 and 2 present some results that explain how the posteriors of our estimation process adapt to upcoming information. We plot the prior distribution and the posterior distribution including up to t periods in our Bayesian estimation framework. Essentially, what can

be deduced from the figures is that the posterior shifts away from the prior and centers around the population value, i.e., the information contained in the data, after just five data points have been observed. This happens for all parameters in the model. Furthermore, these figures show directly how including more data points in the estimation process increases the accuracy by reducing the width of the posterior. We conclude by noting that our model has desirable properties, such as rapid adaptation to upcoming information, an ability to make accurate forecasts with few data points, and sensible estimation results. It is also important to note that the results of our exercises remain fairly constant even when we vary the values of previous market parameters and the scaling factor of the variance, κ , over a relatively wide range, which provides evidence of robustness (Tables D.1 and D.2 in the appendix examine the effect of κ being set to different multiples of the variance in sales σS2 ). 4. Application results As a way of testing the performance of our proposed method on real world data, we consider three applications that cover a wide range of product types and markets, as well as highlighting different aspects of the methodology.

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Table 4 Comparison of one-step-ahead mean squared errors (MSE) and mean absolute errors (MAE) between Bayesian methodologies with different amounts of information without estimating the short-run market. t=0

t=5

t=8

t=9

t = 10

t = 15

t = 20

t = 25

23.18 4.82

0.54 0.74

9.21 3.04

4.23 2.06

0.52 0.72

2.52 1.59

0.01 0.10

24.27 4.93

1.43 1.20

8.37 2.89

6.06 2.46

0.53 0.73

2.52 1.59

0.01 0.10

35.57 5.97

1.04E−03 0.03

13.82 3.72

0.01 0.11

24.18 4.92

17.53 4.19

18.43 4.29

24.55 4.95

0.73 0.85

10.71 3.27

0.03 0.18

24.05 4.90

17.52 4.19

18.43 4.29

Noncumulative sales Scaled-Beta2 MSE MAE

4.26E+04 206.41

Inverse gamma MSE MAE

4.30E+04 207.32

Cumulative sales Scaled-Beta2 MSE MAE

4.26E+04 206.41

Inverse gamma MSE MAE

4.30E+04 207.32

Notes: The models are fitted on each subsample up to t. Scaled-Beta2 has a0 = b0 = 1 and κ0 = 2000. The Inverse Gamma prior has both parameters equal to 0.001. Standard deviations of the chains are given in parentheses.

Fig. 1. Adaptation of the posterior for p to upcoming information. Notes: Bayesian estimation with informative prior. The posterior is computed using different amounts of incoming information.

The first application predicts sales of room air conditioners using information on clothes dryers; the second uses iPod sales to forecast those of BlackBerry handheld devices; and the third focuses on the Colombian gas market, using similar established markets instead of products as a basis for the priors used in sales forecasting. We present estimation results and similar forecast performance measures as we did in our simulation exercise. Our third application also looks at possible alternative scenarios for the paths of the market effort variables and their impacts on performance.

4.1. Room air conditioner We first apply our proposal to the data provided in Table 7 of Bass et al. (1994). The authors collected data on total sales, prices, and advertising for three different products: room air conditioners, color TVs, and clothes dryers for 13 years. Both Bayus (1993) and Jiang et al. (2006) classify room air conditioners and clothes dryers as falling into the same product group (home appliances). Bayus (1993) performs a discriminant analysis and finds that these products are also classified together empirically.

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Fig. 2. Adaptation of posterior for m to upcoming information. Notes: Bayesian estimation with an informative prior. The posterior is computed using different amounts of incoming information. Table 5 Descriptive statistics: room air conditioners.

Table 6 Reference parameter estimates: room air conditioners.

Room air conditioners

Clothes dryer

Variable

Mean

Std. Dev.

Min

Max

p

q

βprice

βadvertising

Sales Cum. sales Price Advertising

1109.077 5674.615 324.154 6.506

652.568 5098.302 48.775 5.694

96 96 259 0

1828 14 418 410 16.785

0.0118 (0.003)

0.2959 (0.0275)

−0.8117

0.6583 (0.2670)

Variable

Mean

Std. Dev.

Min

Max

Sales Cum. sales Price Advertising

965.692 5315.308 214 4.058

465.416 4336.768 17.459 2.673

106 106 185 0

1523 12 554 245 8.657

(0.567)

Notes: This is a replication of the GBM estimates from Bass et al. (1994) via NLS. Standard errors are given in parentheses.

Clothes dryers

Notes: Based on data from Bass et al. (1994).

This means that these products share similar markets and therefore form a perfect setting for our Bayesian method. Table 5 presents the descriptive statistics for the variables and products mentioned above. Times series data are available for thirteen periods for both products. Sales are measured in thousands of units, prices are measured as average prices in dollars, and advertising is measured in millions of dollars. The actual variable that is used as advertising in the estimation process is the product of Simon’s operationalization, which amounts to taking the actual value when the change in advertising is positive and the last value for which there was a positive change otherwise. Our estimation procedure is as follows: first, we estimate the parameters of the GBM for clothes dryers via

nonlinear least squares (see Table 6). Then, as in an empirical Bayes setting (guessing by analogy), we use these estimates as hyperparameters in prior densities for our Bayesian estimation of the GBM for room air conditioners. This approach allows us to avoid using the same information twice, which is a criticism of empirical Bayes methods. Throughout both exercises, we assume that the short term market is fixed.5 The initial values for the NLS and Metropolis–Hastings algorithms are drawn randomly from uniform distributions that fulfill parameter restrictions. As before, we follow Fúquene et al. (2014) and take a0 = b0 = 1 as the hyperparameters for the Scaled-Beta2 prior, and set κ = σS2 , the variance of the sales data. We set the location vector for the multivariate t-prior equal to the point estimates of the NLS procedure from clothes dryers using the full sample; the scale matrix equals the asymptotic covariance matrix, and there are three degrees of freedom. Given that the sales peak for room air conditioners is around t = 8, we will estimate 5 The short-term market is set at 18,475 for room air conditioners and 15,716 for clothes dyers.

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Table 7 Bayesian, NLS and ML estimation: room air conditioners. t=0

t=3

t=8

t = 10

t = 13

0.005 (0.001) 0.332 (0.038) −2.559 (1.392) 0.624 (0.231)

0.005 (0.001) 0.335 (0.020) −1.274 (0.552) 0.593 (0.224)

0.005 (0.001) 0.353 (0.016) −1.092 (0.466) 0.539 (0.211)

0.004 (0.001) 0.276 (0.041) −5.253 (2.714) 0.963 (0.463)

0.005 (0.002) 0.329 (0.024) −1.614 (0.806) 0.641 (0.300)

0.005 (0.001) 0.353 (0.018) −1.219 (0.620) 0.527 (0.245)

0.004 (0.001) 0.282 (0.036) −5.026 (1.788) 0.920 (0.306)

0.005 (0.002) 0.332 (0.024) −1.606 (0.779) 0.654 (0.279)

0.005 (0.002) 0.354 (0.018) −1.236 (0.637) 0.556 (0.256)

0.004 (1.077E−04) 0.285 (0.013) −5.406 (0.573) 0.993 (0.094)

0.004 (1.249E−04) 0.333 (0.006) −2.159 (0.201) 0.757 (0.052)

0.004 (1.270E−04) 0.355 (0.005) −1.720 (0.167) 0.601 (0.039)

Bayesian estimation: informative prior p q

β1 β2

0.006 (0.106) 0.295 (0.280) −0.853 (1.369) 0.631 (0.854)

0.005 (0.001) 0.244 (0.153) −1.052 (1.129) 0.442 (0.511)

Bayesian estimation: non-informative prior p

–

–

q

–

–

β1

–

–

β2

–

–

NLS estimation p

–

–

q

–

–

β1

–

–

β2

–

–

ML estimation p

–

q

–

β1

–

β2

–

0.003 (3.175E−04) 0.787 (0.246) −0.409 (1.845) −0.571 (0.241)

Notes: The models are fitted on each subsample up to t. Informative priors use the market estimates of dryers as hyperparameters, while non-informative priors are centered at zero with a high variance. NLS and MLE include appropriate parameter restrictions. Standard errors are given in parentheses (standard deviation of chains for Bayesian methods and asymptotic for frequentist). ‘‘–’’ represents unavailable estimation results.

the model at periods t = 0, 3, 8, 10, 13. Takeoff and saturation measures were not available for this application, due to the particular shape of the sales curve for this product, but we compute these for the two remaining applications whenever possible. Table 7 presents the estimation results for the Bayesian estimation, running one chain with 400,000 iterations, a burn-in period of 200,000 and a thinning parameter equal to 50, and the frequentist (NLS and ML) techniques. We observe that the signs of the estimates are consistent with the theory. In particular, price increases have a negative effect on sales, while advertisement increases have a positive effect. As more information is included, the estimation of all parameters becomes more accurate and the standard error tends to decrease. We see that it is not possible to obtain estimates before the sales peak using the NLS approach, due to problems of convergence in the optimization process. All of the coefficients are statistically significant, and all of the chains converge to a stationary distribution with desirable properties according to several convergence diagnostics (available upon request).

Table D.3 in the appendix examines the consequences of misspecifying this value. In general, both under- and over-specifying m generates a loss in precision, as can be seen from the increased standard errors, particularly for the marketing effort coefficients. While the innovation parameter appears to be consistent across all scenarios of the short-term market, the imitation parameter varies considerably, which provides evidence of the importance of specifying this quantity correctly. If a manager is uncertain of the short-term market for their product and historical data or information for similar markets are available that could be used as a way of calibrating the prior distributions, the manager could estimate m directly and check both scenarios. Table 8 presents mean squared and mean absolute forecast errors for our Bayesian approach and the NLS and ML methods. We observe that the errors generally decrease with t after t = 3. The Bayesian prediction errors dominate, particularly around the sales peak. By the time that all of the data have been used in the forecasting

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Table 8 Root mean squared errors (RMSE) and mean absolute errors (MAE) with different amounts of information: room air conditioners. t=0

t=3

t=8

t = 10

t = 13

Noncumulative sales

220.404 190

563.055 458

139.742 101

126.404 110

114.310 92

Bayesian estimation: non-informative prior RMSE MAE

– –

– –

239.173 155

129.884 114

113.964 89

– –

230.364 151

127.263 111

113.789 89

830.254 600

239.455 155

127.824 102

119.549 81

201.995 178

120.037 89

201.151 175

118.055 88

NLS estimation RMSE MAE

– –

ML estimation RMSE MAE

– –

Cumulative sales

886.645 643

3301.288 2452

136.304 117

Bayesian estimation: non-informative prior RMSE MAE

– –

– –

414.223 263

NLS estimation RMSE MAE

– –

– –

342.616 225

184.732 161

111.458 85

2614.999 1779

348.493 212

137.331 95

106.193 94

ML estimation RMSE MAE

– –

t=8

t = 10

t = 13

MAE RMSE AIC BIC

458.000 563.055 49.296 45.691

101.000 139.742 103.672 103.990

110.000 126.404 127.816 129.026

92.000 114.310 164.556 166.816

BM: Bayesian estimation with informative prior MAE RMSE AIC BIC

315.104 430.696 45.688 42.984

170.612 202.426 107.602 107.840

153.768 166.521 131.328 132.236

144.339 162.526 171.706 173.401

264.861 335.054 115.664 115.903

151.225 186.418 133.586 134.493

156.276 189.790 175.738 177.433

BM: Hong et al. (2016) MAE RMSE AIC BIC

404.609 518.294 46.799 44.095

BM with integration constant: Kim and Hong (2015)

Bayesian estimation: informative prior RMSE MAE

t=3 Noncumulative sales

GBM: Bayesian estimation with informative prior

Bayesian estimation: informative prior RMSE MAE

Table 9 Forecast performances of competing models: room air conditioners.

MAE RMSE AIC BIC

1043.900 1257.800 54.119 50.513

391.299 590.600 126.734 127.052

286.274 423.723 152.007 153.218

152.904 179.282 176.257 178.517

GGM: Guseo et al. (2017) and Guseo and Guidolin (2009) MAE RMSE AIC BIC

976.932 1185.100 55.761 51.254

126.203 149.367 106.738 107.135

122.082 141.875 132.125 133.638

99.597 117.258 167.218 170.043

Cumulative sales

Notes: The models are fitted on each subsample up to t. Informative priors use the market estimates of dryers as hyperparameters, while non-informative priors are centered at zero with a high variance. NLS and MLE include appropriate parameter restrictions. Minimum values are shown in bold. ‘‘–’’ represents unavailable estimation results.

process, the NLS is generally better than the Bayesian approach. In addition, we also compare the predictive performances of different model specifications, namely the GBM and BM using informative priors, the Bass model using Hong et al.’s (2016) hybrid approach of NLS and OLS (BM HON), the Bass model with integration constant (BMIC) proposed by Kim and Hong (2015),6 and Guseo et al.’s (2017) model (GMM). All optimization algorithms for estimating these models had hyperparameters from our prior distributions as initial points. Table 9 shows various different predictive performance criteria, specifically the mean absolute error, root mean squared error (RMSE), Akaike information criterion (AIC) and Bayesian information criterion (BIC). We present the AIC and BIC because we are computing results for models with different levels of complexity (dimensions of parameter space). We observe in this table that the best predictive performance (noncumulative and cumulative 6 We do not estimate the virtual Bass model (VBM) proposed by Jiang et al. (2006) due to its requirement for information about the launch time. In addition, it seems that the BMIC outperforms the VBM.

GBM: Bayesian estimation with informative prior MAE RMSE AIC BIC

2452 3301.290 59.908 56.303

117.000 136.304 103.274 103.592

178.056 201.995 137.191 138.401

89.012 120.037 165.827 168.087

BM: Bayesian estimation with informative prior MAE RMSE AIC BIC

554.753 2541.597 56.339 53.635

1961.263 637.627 125.960 126.198

500.655 420.422 149.851 150.759

256.777 297.338 187.411 189.106

648.251 864.486 130.830 131.068

276.119 320.366 144.415 145.323

267.917 320.230 189.339 191.034

BM: Hong et al. (2016) MAE RMSE AIC BIC

2465.500 3132.200 57.593 54.889

BM with integration constant: Kim and Hong (2015) MAE RMSE AIC BIC

4963.000 6866.400 64.302 60.697

806.898 1529.000 141.953 142.271

434.931 857.313 166.102 167.312

156.863 181.645 176.598 178.858

GGM: Guseo et al. (2017) and Guseo and Guidolin (2009) MAE RMSE AIC BIC

4585.700 6383.700 65.865 61.358

249.808 351.528 120.432 120.829

135.360 200.128 139.005 140.518

61.738 74.299 155.355 158.180

Notes: The models are fitted on each subsample up to t. Informative priors use the market estimates of dryers as hyperparameters. BMIC and GGM are estimated by NLS. Minimum values are shown in bold.

sales) is achieved using informative Bayesian approaches (BM and GBM), except when using all sample information (t = 13) in cumulative sales, when the GGM model achieves the best predictive performance.

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Fig. 3. Forecasts of noncumulative sales with different amounts of data: room air conditioners. Notes: The forecasts are from Bayesian estimation with informative priors. Parameter point estimates are used to compute full forecasts.

One of the most important results for our approach is the fact that it allows forecasts even when there are few or no data points (pre-launch). Figs. 3 and 4 show noncumulative and cumulative forecasts, respectively, with varying amounts of information. As expected, having more data points tends to improve the sales forecast; it appears that the cumulative forecast tends to be more accurate than the noncumulative forecast when using more data points. Lastly, Fig. 5 presents the one-step-ahead forecast for cumulative sales. When most useful, namely at the introduction of a new product when there are no historical data, the one-step-ahead forecast provides a fair approximation of the actual cumulative sales. As more periods pass, the forecast appears to adapt well to the shape of the distribution for cumulative room air conditioner sales. 4.2. BlackBerry handheld devices Our second application relates to BlackBerry handheld devices. We collected annual data on units sold (in millions) and real prices (in US dollars) from 2005 to 2013,7 and used information on iPod sales between 2002 and 2014 to obtain the hyperparameters of our prior distributions.8 This choice was based on data availability, in particular average prices, and on the fact that both are electronic devices in contemporary time periods that exhibit similar downward trends in prices, as well as sales peaks at the seventh year after launching. In particular, 7 https://barefigur.es/companies/blackberry/. 8 https://barefigur.es/companies/apple/ipod/.

Table 10 Descriptive statistics: BlackBerry handheld devices. BlackBerry handheld devices Variable

Mean

Std. Dev.

Min

Max

Sales Cum. sales Price

25.591 100.318 423.571

17.664 89.258 59.782

3.640 3.640 357.139

52.800 230.320 524.725

iPod Variable

Mean

Std. Dev.

Min

Max

Sales Cum. sales Price

30.891 196.971 244.928

20.518 156.280 286.632

1.519 1.519 119.804

56.001 401.586 1171.824

Notes: Based on Bare Figures for BlackBerry (https://barefigur. es/companies/blackberry/) and Apple (https://barefigur.es/companies/ apple/). Prices are given in 2002 US dollars.

BlackBerry’s sales peak was reached in 2011, and iPod’s in 2008 (see Table 10 for descriptive statistics). While there are not enough data after the peak to provide a satisfactory measure of the saturation point for the sales of this product, all of our methods agree on a population takeoff point around the fourth or fifth year. We acknowledge that these characteristics are not known prior to product launch; however, this application highlights an interesting feature of our proposal, as we will show later. We used the hyperparameters that were obtained using NLS (see Table 11) to forecast cumulative sales of BlackBerry devices. We ran three parallel chains, each one with 50,000 iterations, 25,000 as burn-in, and a thinning parameter equal to 25. This yields an effective posterior chain length of 3000.

Please cite this article as: A. Ramírez-Hassan and S. Montoya-Blandón, Forecasting from others’ experience: Bayesian estimation of the generalized Bass model. International Journal of Forecasting (2019), https://doi.org/10.1016/j.ijforecast.2019.05.016.

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Fig. 4. Forecasts of cumulative sales with different amounts of data: room air conditioners. Notes: The forecasts are from Bayesian estimation with informative priors. Parameter point estimates are used to compute full forecasts.

Fig. 5. One-step-ahead forecasts of cumulative sales: room air conditioners. Notes: The forecasts are from Bayesian estimation with informative priors. Parameter point estimates are used to compute one-step-ahead forecasts.

As can be seen from Fig. 6, our choice of hyperparameter was not good at all, because a model that forecasts negative sales is unacceptable, despite the fact that we apparently had convincing arguments for our choice.

However, this illustrates another interesting characteristic of our Bayesian proposal: the fast adaptation of the algorithm to sample information. As Figs. 7 and 8 show, the Bayesian estimation reaches a relatively good forecasting

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Fig. 6. Forecasts of noncumulative sales with prior distribution (no data/pre-launch): BlackBerry handheld devices. Notes: The forecasts are from Bayesian estimation with informative priors but no data for constructing the posterior. Hyperparameters are used to compute full forecasts. Table 11 Reference parameter estimates: BlackBerry handheld devices. iPod p

q

βprice

Short-term market

0.0037 (0.0030)

0.5320 (0.0398)

−0.9898

424.2602 (23.6861)

(0.6801)

Notes: Based on Bare Figures for Apple (https://barefigur.es/companies/ apple/). The estimation of GBM is performed via NLS. Standard errors are given in parentheses.

performance using just two sample points, even in the presence of bad prior information. Obviously, using good prior information would allow even better sales forecasts to be obtained, as was observed in the simulation exercises. We compare different estimation strategies for the GBM, namely informative and non-informative Bayesian, and NLS.9 We always obtained parameter estimates using an informative Bayesian approach, whereas noninformative Bayesian and NLS estimates were obtained using only t = 8 and t = 6, 8, respectively. This implies that informative priors help to avoid convergence issues. Table 12 shows parameter estimates. In general, we obtain parameter estimates in the theoretical range. Specifically, we observe an upward trend in the imitation parameter as the sample size increases using the informative Bayesian approach. We get an imitation parameter of 0.008 using a sample size of eight. This is the same 9 We did not obtain ML estimates due to the restriction of defining the long-term potential market in this method.

value that Lilien, Rangaswamy, and Van Den Bulte (2000) showed for cellular phones using data from 1986 to 1996 (see their Tables 12.1 and 12.2, pp. 300, 302). We observe that the standard errors of the Bayesian approaches are lower than the standard errors of NLS. Table 13 presents the forecasting errors. In this particular application, we observe that the forecast errors of noncumulative sales using NLS, whenever it is possible to apply it (t = 6, 8), are lower than those associated with the Bayesian approaches. On the other hand, the lowest forecast errors of cumulative sales at t = 8 occur using the non-informative Bayesian approach. Otherwise, the informative Bayesian estimation of the GBM achieves the best predictive performance. In addition, we also compare the predictive performances of different model specifications in Table 14. We observe that informative Bayesian procedures achieve the best performances at t = 2, 4, that is, when little sample information is available. On the other hand, BMIC performs the best at t = 6, 8, that is, when the sales peak is available. Similarly, we see that the Bayesian approaches provide good approximations regarding the managerial metrics of peak and takeoff, again with small amounts of information at t = 2, 4, followed closely by the BMIC. As expected, the frequentist approaches catch up when enough information is supplied. Fig. 9 shows another interesting feature of our proposal: the simple estimation of predictive credible intervals. This allows us to obtain alternative forecast scenarios just by using the posterior chains, without extra computational effort, and taking into account estimation error. This task is complicated when using frequentist

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Fig. 7. Forecasts of noncumulative sales with different amounts of data: BlackBerry handheld devices. Notes: The forecasts are from Bayesian estimation with informative priors. Parameter point estimates are used to compute full forecasts.

Fig. 8. Forecasts of cumulative sales with different amounts of data: BlackBerry handheld devices. Notes: The forecasts are from Bayesian estimation with informative priors. Parameter point estimates are used to compute full forecasts.

methods due to the non-linearity of the GBM diffusion equations (see Eqs. (4) to (6)), which prohibits exact analytic solutions of the standard deviation associated with

the prediction. As a consequence, its calculation requires asymptotic approximations, whenever they are possible, or imposes an extra computational burden.

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Fig. 9. 95% one-step-ahead credible predictive interval of cumulative sales: BlackBerry handheld devices. Notes: Bayesian estimation with informative priors is used to construct the chains and predictive intervals.

Table 12 Bayesian and NLS estimation: BlackBerry handheld devices. t=2

t=4

t=6

t=8

Table 13 Mean absolute errors (MAE) and root mean squared errors (RMSE): BlackBerry handheld devices. t=2

Bayesian estimation: informative prior p q

βprice m

0.005 (3.19E−05) 0.528 (0.001) −0.844 (0.017) 426.174 (0.680)

0.006 (2.91E−05) 0.549 (0.001) −1.177 (0.013) 426.948 (0.682)

0.007 (2.26E−05) 0.567 (0.001) −0.836 (0.011) 427.354 (0.611)

0.008 (5.90E−05) 0.549 (0.001) −1.139 (0.014) 367.326 (1.187)

Bayesian estimation: non-informative prior p

–

–

–

q

–

–

–

βprice

–

–

–

m

–

–

–

–

–

q

–

–

βprice

–

–

m

–

–

0.007 (5.17E−04) 0.687 (0.039) −0.410 (0.260) 295.160 (31.749)

t=6

t=8

9.031 16.048

5.870 9.762

Bayesian estimation: informative prior MAE RMSE

8.521 10.856

9.305 15.458

Bayesian estimation: non-informative prior MAE RMSE

– –

– –

– –

1.596 1.902

– –

– –

2.945 5.633

1.135 1.235

15.186 29.285

7.156 11.219

NLS estimation 0.008 (1.6E−04) 0.701 (0.003) −1.293 (0.033) 263.870 (0.925)

NLS estimation p

t=4

Noncumulative sales

0.005 (8.68E−04) 0.747 (0.042) −1.229 (0.494) 250.313 (8.108)

Notes: The models are fitted on each subsample up to t. Informative priors use the market estimates of iPods as hyperparameters, while non-informative priors are centered at zero with a high variance. NLS includes appropriate parameter restrictions. Standard errors are given in parentheses (standard deviation of chains for Bayesian methods and asymptotic for frequentist). ‘‘–’’ represents unavailable estimation results.

MAE RMSE

Cumulative sales Bayesian estimation: informative prior MAE RMSE

15.420 20.928

12.073 22.670

Bayesian estimation: non-informative prior MAE RMSE

– –

– –

– –

1.833 1.596

– –

– –

4.282 9.122

2.115 2.257

NLS estimation MAE RMSE

Notes: The models are fitted on each subsample up to t. Informative priors use the market estimates of iPods as hyperparameters, while non-informative priors are centered at zero with a high variance. NLS includes appropriate parameter restrictions. Minimum values are shown in bold. ‘‘–’’ represents unavailable estimation results.

We observe from Fig. 9 that the 95% one-step-ahead credible predictive interval of cumulative sales gives sensible results, always embracing the actual cumulative sales.

Please cite this article as: A. Ramírez-Hassan and S. Montoya-Blandón, Forecasting from others’ experience: Bayesian estimation of the generalized Bass model. International Journal of Forecasting (2019), https://doi.org/10.1016/j.ijforecast.2019.05.016.

16

A. Ramírez-Hassan and S. Montoya-Blandón / International Journal of Forecasting xxx (xxxx) xxx

Table 14 Forecast performances of competing models: BlackBerry handheld devices. t=2

t=4

t=6

t=8

Noncumulative sales GBM: informative prior MAE RMSE AIC BIC Peak Takeoff

8.521 10.856 18.925 13.698 8 6

9.305 15.458 35.450 32.995 8 5

9.031 16.048 52.058 51.225 8 5

5.870 9.762 61.091 61.409 7 –

15.778 9.018 29.139 27.298 8 5

16.763 9.133 43.293 42.669 8 5

12.919 8.332 56.557 56.796 8 5

18.505 24.993 37.294 35.453 9 –

15.317 22.003 53.845 53.220 9 –

7.507 9.635 58.882 59.120 8 5

21.343 33.236 41.574 39.119 8 6

2.877 4.928 37.890 37.057 7 5

1.887 2.573 39.757 40.074 7 4

87.883 154.005 55.841 52.773 9 –

3.186 5.755 41.751 40.710 7 5

1.924 2.636 42.144 42.541 7 4

BM: informative prior MAE RMSE AIC BIC Peak Takeoff

15.384 11.618 17.197 13.276 9 6

BM HON: Hong et al. (2016) MAE RMSE AIC BIC Peak Takeoff

15.068 19.110 19.187 15.267 9 –

BMIC: Kim and Hong (2015) MAE RMSE AIC BIC Peak Takeoff

10.355 13.270 19.728 14.501 8 6

GGM: Guseo et al. (2017) MAE RMSE AIC BIC Peak Takeoff

11.304 15.146 22.257 15.723 9 6

Cumulative sales GBM: informative prior MAE RMSE AIC BIC

15.420 20.928 21.551 16.323

12.073 22.670 38.514 36.059

15.186 29.285 59.275 58.442

7.156 11.219 63.317 63.635

21.070 9.018 29.139 27.298

29.101 9.133 43.293 42.669

13.591 8.332 56.557 56.796

33.318 45.014 42.001 40.160

22.767 30.646 57.821 57.196

8.313 9.713 59.011 59.249

46.179 79.913 48.593 46.138

3.419 6.875 41.885 41.052

1.673 2.281 37.829 38.147

BM: informative prior MAE RMSE AIC BIC

26.943 11.618 17.197 13.276

BM HON: Hong et al. (2016) MAE RMSE AIC BIC

54.303 70.724 24.421 20.501

BMIC: Kim and Hong (2015) MAE RMSE AIC BIC

20.417 27.624 22.661 17.434

(continued on next page)

Table 14 (continued). t=2

t=4

t=6

t=8

160.826 304.864 61.304 58.236

4.114 8.573 46.534 45.493

1.746 2.358 40.361 40.758

GGM: Guseo et al. (2017) MAE RMSE AIC BIC

38.454 51.011 27.114 20.580

Notes: The models are fitted on each subsample up to t. Informative priors use the market estimates of iPods as hyperparameters. BMIC and GGM are estimated by NLS. Minimum values are shown in bold. ‘‘–’’ represents unavailable estimation results. Table 15 Descriptive statistics: compressed natural gas. Objective market: Santa Rosa Variable

Mean

Std. Dev.

Min

Max

Sales Cum. sales CNG price LPG price

278 2099 0.476 0.922

308 1128 0.019 0.057

31 134 0.460 0.828

1055 3055 0.510 1.020

Reference market: San Pedro Variable

Mean

Std. Dev.

Min

Max

Sales Cum. sales CNG price LPG price

176 1333 0.421 0.921

178 672 0.089 0.088

10 211 0.213 0.796

505 1941 0.492 1.117

Reference market: Don Matías Variable

Mean

Std. Dev.

Min

Max

Sales Cum. sales CNG price LPG price

162 838 0.471 0.943

127 696 0.023 0.086

1 1 0.434 0.817

432 1783 0.507 1.141

Notes: Based on Sistema Único de Información (www.sui.gov.co).

4.3. Compressed natural gas Our third application concerns the forecast of early residential adopters of compressed natural gas in a small municipality in the province of Antioquia (Colombia). This data set comprises monthly data from October 2011 to August 2012. We build the market effort variables using compressed natural gas (CNG) and liquid petroleum gas (LPG) real prices (December 2008), both expressed in US$/M 3 , taking into account the calorific power of each so as to make them comparable. Previous to the introduction of CNG, LPG was the predominant fuel used in objective households, and we expect that its price should have a positive effect on the adoption rate a priori, whereas the CNG price should have a negative effect. Table 15 shows the descriptive statistics associated with three markets: Santa Rosa, our forecast objective, and two reference markets, San Pedro and Don Matías. These three municipalities are located in the rural area of the province. We can see from the table that the compressed natural gas and liquid petroleum gas prices exhibit similar patterns in the three markets. This is due to government regulation restrictions and competitive pressure. In particular, compressed natural gas is provided by Empresas Públicas de Medellín, one of the most important utility companies in Colombia, which holds the monopoly

Please cite this article as: A. Ramírez-Hassan and S. Montoya-Blandón, Forecasting from others’ experience: Bayesian estimation of the generalized Bass model. International Journal of Forecasting (2019), https://doi.org/10.1016/j.ijforecast.2019.05.016.

A. Ramírez-Hassan and S. Montoya-Blandón / International Journal of Forecasting xxx (xxxx) xxx Table 16 Reference parameter estimates: compressed natural gas. San Pedro p

q

βCNG

βLPG

Short-term market

0.031

0.869

−0.641

3.167

2070

p

q

βCNG

βLPG

Short-term market

0.011

0.644

−12.220

5.693

1852

Don Matías

Notes: Based on Sistema Único de Información (www.sui.gov.co). The estimation of GBM is via NLS.

on this type of fuel in the province. As all three municipalities are part of this company’s market, the compressed natural gas price is driven by the same rules, with only minor modifications that are associated with known technical components. In addition, liquid petroleum gas production and transport at a national level is likewise a strictly-regulated monopoly, whereas the distribution and commercialization at regional level are regulated by a maximum price. This implies a price-cap regulation framework that allows competition at the local level. For both products, the regulation period is 5 years. This establishes an excellent framework that facilitates forecasts of the market effort in our application due to government regulations establishing a maximum price, and companies having some degree of freedom to drive prices given this limit. In our objective market, the sales peak is reached after four periods, leaving insufficient data for computing the takeoff point, but allowing the population saturation to be computed as occurring at around the fifth or sixth period. Table 16 presents the estimates associated with the innovation, imitation, adoption price elasticities, and shortterm markets of the two reference markets. We estimate these parameters using NLS and historical records of the two previous markets for the municipalities. Then, we use the results as inputs to calculate the hyperparameters of our prior distributions for the informative Bayesian framework (guessing by analogy). This application shows the flexibility of the Bayesian framework by using beta prior distributions for the innovation and imitation parameters. In addition, we use truncated Student’s t distributions with three degrees of freedom for the price adoption elasticities and the shortterm market. In particular, we impose a negative support for the compressed natural gas elasticity, and positive supports for the liquid petroleum gas elasticity and the short-term market. These restrictions are due to theoretical considerations, and apply to both the informative and non-informative Bayesian estimations. In the case of informative Bayesian estimations, p ∼ B(2.077, 96.419), q ∼ B(4.781, 1.539), βCNG ∼ t(−∞,0) (−6.430, 67.032, 3), βLPG ∼ t(0,∞) (4.429, 3.190, 3), βm ∼ t(0,∞) (1961.082, 154.0411, 3) and σ 2 ∼ SBeta2(1, 1, 5). The hyperparameters of )p ∼ B(α, β ) were (calculated ( ) using α =

1−E(p) Var(p)

−

1 E(p)

(E(p))2 and β = α

1 E(p)

−1 ,

where E(p) and Var(p) are the mean and variance obtained from the reference markets. Similar formulas were used for the imitation parameters. In addition, the mean

17

and variance from the reference markets were used to calculate the hyperparameters of the other prior distributions, with the exception of the prior for the model variance, which is based on the work of Fúquene et al. (2014). This prior setting is used in conjunction with the likelihood (Eq. (8)) to perform the Bayesian posterior analysis. Table 17 presents the posterior estimates at different sample sizes, running three chains for each parameter with 200,000 iterations, a burn-in of 100,000, and a thinning parameter of 50. Thus, the effective length of each posterior chain is 6000. This setting generates convergence of the chains. At t = 0, that is, without sample information, we obtain results using just prior information from reference markets in our informative Bayesian setting. In particular, we implement exactly the same procedure for all parameters as that described above, except for the short-term market. We use a beta prior distribution to estimate this parameter. We calculate the ratio between the short-term and long-term markets from reference markets, and apply an empirical Bayes approach to get hyperparameters. Then, we multiply these draws by the long-term market of the objective market. We use electricity users as long-term markets. Regarding non-informative Bayesian estimations, we use B(1, 1) distributions (that is, uniform distributions) for the innovation and imitation parameters, and a vague truncated Student’s t distribution for the remaining parameters according to their theoretical values. All Bayesian parameter estimates are significant at conventional levels. On the other hand, NLS estimation had convergence problems at all sample sizes, despite using evolutionary search algorithms with derivative-based methods that do not require starting values (the PORT algorithm failed in this application). Even when imposing the same parameter restrictions as in the Bayesian frameworks, the parameter estimates are not located in the theoretical range. In addition, the NLS estimates have a huge level of variability depending on the sample size, and it was not possible to obtain standard error estimates at t = 3. We produce forecasts using each possible posterior value of the chains, and thus obtain 6000 forecasts for each period. This allows us to create credible intervals or any inferential quantity of interest, which is a difficult task using traditional approaches. We obtain a point forecast by calculating the mean values at each time period, which minimizes a quadratic loss function. Table 18 shows forecast errors (noncumulative and cumulative) using our Bayesian proposals, and NLS at different sample sizes. We observe that the informative Bayesian estimation has the lowest errors, except at t = 0 and t = 5, when non-informative Bayesian estimation is the best. It seems again that there is a very rapid process of sample information updating in the Bayesian procedures. We also compare the predictive performances of different model specifications. We see in Table 19 that our Bayesian proposals (informative and non-informative) exhibit the best predictive performances regarding noncumulative sales, except at t = 5. Again, it seems that using prior information from reference markets (guessing by

Please cite this article as: A. Ramírez-Hassan and S. Montoya-Blandón, Forecasting from others’ experience: Bayesian estimation of the generalized Bass model. International Journal of Forecasting (2019), https://doi.org/10.1016/j.ijforecast.2019.05.016.

18

A. Ramírez-Hassan and S. Montoya-Blandón / International Journal of Forecasting xxx (xxxx) xxx Table 17 Bayesian and NLS estimation: compressed natural gas. t=0

t=3

t=5

t=7

t=9

t = 11

0.016 (4.54E−05) 0.907 (0.001) −4.708 (0.045) 24.639 (0.112) 2222.836 (3.111)

0.040 (1.15E−04) 0.949 (0.001) −4.226 (0.025) 7.846 (0.019) 2847.278 (3.011)

0.047 (1.01E−04) 0.963 (4.17E−04) −5.194 (0.021) 7.772 (0.014) 2807.634 (1.674)

0.047 (1.23E−04) 0.956 (4.52E−04) −5.003 (0.026) 7.713 (0.018) 2812.296 (1.883)

0.296 (0.003) 0.469 (0.003) −14.843 (0.087) 11.698 (0.114) 1771.967 (11.927)

0.761 (0.001) 0.504 (0.003) −16.726 (0.016) 1.962 (0.0143) 2306.604 (3.9143)

0.143 (0.002) 0.761 (0.003) −10.203 (0.093) 9.285 (0.055) 2807.145 (4.532)

0.091 (0.001) 0.889 (0.002) −8.011 (0.057) 8.507 (0.0308) 2778.342 (2.379)

0.107 (1.78E−04) 0.833 (3.09E−04) −8.878 (0.065) 8.624 (0.029) 2788.807 (2.390)

2.62E−05 – 0.906 – −1.806 – 94.876 – 5893.141 –

0.003 (1.11E−05) 0.325 (0.006) −1.34E−14 (0.163) 27.225 (0.707) 9842.058 (128.544)

0.011 (6.73E−05) 1.245 (0.002) −3.936 (0.044) −2.269 (0.027) 2672.426 (1.582)

0.017 (8.97E−05) 1.047 (0.001) −6.952 (0.048) −0.301 (0.030) 2816.768 (1.632)

0.016 (8.94E−05) 1.051 (0.001) −7.067 (0.049) −0.228 (0.030) 2792.053 (1.602)

Bayesian estimation: informative prior p q

βCNG βLPG m

0.021 (1.81E−04) 0.754 (0.002) −11.197 (0.131) 4.793 (0.033) 3222.362 (5.856)

0.053 (2.24E−04) 0.798 (0.002) −3.360 (0.036) 8.085 (0.077) 2060.048 (3.195)

Bayesian estimation: non-informative prior p q

βCNG βLPG m

0.500 (0.003) 0.500 (0.003) −36.503 (0.517) 36.677 (0.549) 2726.938 (20.390)

NLS estimation p

–

q

–

βCNG

–

βLPG

–

m

–

Notes: The models are fitted on each subsample up to t. Informative priors use the market estimates of municipalities as hyperparameters, while non-informative priors are centered at zero with a high variance. NLS includes appropriate parameter restrictions. Standard errors are given in parentheses (standard deviation of chains for Bayesian methods and asymptotic for frequentist). ‘‘–’’ represents unavailable estimation results.

analogy) helps to improve the predictive performances. The same conclusion follows in terms of peak and saturation, where the Bayesian methods obtain satisfactory results with small amounts of information, and perform comparably to the best frequentist alternative of GGM as the number of periods increases and the peak is passed. The latter is the best model for evaluating cumulative sales using t = 9 and t = 11 sample information for estimation. Figs. 10 and 11 present the forecast evolution (noncumulative and cumulative) at different sample sizes using the GBM. While there are obvious mistakes when using only prior information, our proposal follows the adoption pattern after three sample points, even before the sales peak, but there is a scale error due to an underestimate of the short-term market (see Table 17). We basically replicate the adoption pattern using a sample size of seven, which is practically the same as we got using 11 sample points. Fig. 12 shows the 95% credible predictive interval using a sample size of 11. As can be seen from this figure, our informative Bayesian proposal performs well. In general, the credible predictive interval embraces the observed sales.

Fig. 13 presents a scenario analysis forecasting noncumulative and cumulative sales of compressed natural gas under three different scenarios of CNG prices, namely base (observed), 5% decrease, and 5% increase. We show forecasts using information from up to three and seven periods for estimation. Posterior distributions can be obtained via CNG,j w Obs LPG π (SNe 1:T |S1:t , P1:T , P1:T ) ∫ ∫ CNG,j w 2 LPG = f (SNe 1:T |θ, σ , P1:T , P1:T )

R++

Θ

CNG LPG 2 × π (θ, σ 2 |SObs 1:t , P1:t , P1:t )dθ dσ

≈

S 1∑

S

CNG,j

w (s) 2(s) f (SNe , P1:T , PLPG 1:T |θ , σ 1:T ),

s=1

where j = {base (obser v ed), 5% decrease, 5% increase}, and θ (s) and σ 2(s) are draws from the posterior distributions. We observe in Fig. 13 that a 5% decrease in the CNG price implies a higher level of the sales peak, a feature that does not depend on the estimation sample available (t = 3 and t = 7). After this period, the scenario with a 5% price increase exhibits more noncumulative sales. As expected, cumulative sales are the same at the final

Please cite this article as: A. Ramírez-Hassan and S. Montoya-Blandón, Forecasting from others’ experience: Bayesian estimation of the generalized Bass model. International Journal of Forecasting (2019), https://doi.org/10.1016/j.ijforecast.2019.05.016.

A. Ramírez-Hassan and S. Montoya-Blandón / International Journal of Forecasting xxx (xxxx) xxx

19

Table 18 Mean absolute errors (MAE) and root mean squared errors (RMSE): compressed natural gas. t=0

t=3

t=5

t=7

t=9

t = 11

119.341 187.332

36.348 41.696

29.665 35.499

31.007 36.749

126.668 185.164

61.606 96.451

37.732 44.881

31.812 38.703

34.400 41.059

1643.625 2191.117

281.453 444.271

56.067 73.137

54.679 63.288

54.474 63.187

471.828 571.312

43.324 48.964

41.226 54.241

42.550 53.337

886.596 980.347

293.852 384.124

57.017 72.313

50.522 70.532

47.171 66.897

1220.155 1669.224

1308.983 1660.980

137.891 181.192

77.450 94.108

78.345 99.000

Noncumulative sales Bayesian estimation: informative prior MAE RMSE

258.629 328.902

102.042 157.780

Bayesian estimation: non-informative prior MAE RMSE

184.857 291.164

NLS estimation MAE RMSE

– –

Cumulative sales Bayesian estimation: informative prior MAE RMSE

486.940 664.254

652.693 719.268

Bayesian estimation: non-informative prior MAE RMSE

269.524 350.605

NLS estimation MAE RMSE

– –

Notes: The models are fitted on each subsample up to t. Informative priors use the market estimates of municipalities as hyperparameters, while non-informative priors are centered at zero with a high variance. NLS includes appropriate parameter restrictions. Minimum values are shown in bold. ‘‘–’’ represents unavailable estimation results.

Fig. 10. Forecasts of noncumulative sales with different amounts of data: Compressed natural gas. Notes: The forecasts are from Bayesian estimation with informative priors. Parameter point estimates are used to compute full forecasts.

period under different scenarios. From a manager’s perspective, decreasing CNG prices accelerates the adoption rate, which improves the present value of the future cash

flow that is associated with new connections to the CNG network, thus leveraging the project in the short-term. However, it is necessary to analyze the income that is

Please cite this article as: A. Ramírez-Hassan and S. Montoya-Blandón, Forecasting from others’ experience: Bayesian estimation of the generalized Bass model. International Journal of Forecasting (2019), https://doi.org/10.1016/j.ijforecast.2019.05.016.

20

A. Ramírez-Hassan and S. Montoya-Blandón / International Journal of Forecasting xxx (xxxx) xxx

Fig. 11. Forecasts of cumulative sales with different amounts of data: compressed natural gas. Notes: The forecasts are from Bayesian estimation with informative priors. Parameter point estimates are used to compute full forecasts.

Fig. 12. 95% credible predictive intervals for noncumulative sales: compressed natural gas. Notes: Bayesian estimation with informative priors is used to construct the chains and predictive intervals.

associated with CNG total consumption, which depends

consumption price elasticity: if CNG is an elastic product,

on the number of connections, as well as the average

a 5% price decrease also improves the income that is

consumption per connection. The latter is affected by the

associated with consumption, making this a profitable

Please cite this article as: A. Ramírez-Hassan and S. Montoya-Blandón, Forecasting from others’ experience: Bayesian estimation of the generalized Bass model. International Journal of Forecasting (2019), https://doi.org/10.1016/j.ijforecast.2019.05.016.

A. Ramírez-Hassan and S. Montoya-Blandón / International Journal of Forecasting xxx (xxxx) xxx Table 19 Forecast performances of competing models: compressed natural gas. t=3

t=5

t=7

t=9

t = 11

Noncumulative sales GBM: Bayesian estimation with informative prior MAE RMSE AIC BIC Peak Saturation

102.042 157.780 43.663 39.156 4 6

119.341 187.332 70.376 68.423 4 7

36.348 41.696 75.847 75.577 4 6

29.665 35.499 94.026 95.012 4 6

31.007 36.749 115.667 117.657 4 6

21

Table 19 (continued). t=3

t=5

t=7

t=9

t = 11

GGM: Guseo et al. (2017) and Guseo and Guidolin (2009) MAE RMSE AIC BIC

1149.300 1460.600 57.015 52.509

247.647 326.662 75.936 73.984

50.853 55.325 79.807 79.536

33.226 41.004 96.621 97.607

32.515 40.887 118.015 120.004

Notes: The models are fitted on each subsample up to t. Informative priors use the market estimates of municipalities as hyperparameters. BMIC and GGM are estimated by NLS. Minimum values are shown in bold. ‘‘–’’ represents unavailable estimation results.

BM: Bayesian estimation with informative prior MAE RMSE AIC BIC Peak Saturation

93.408 189.755 40.770 38.066 4 7

411.838 654.772 78.890 77.718 6 8

121.065 221.368 95.219 95.057 5 7

95.775 187.586 119.991 120.583 4 7

83.372 164.773 144.677 145.871 4 7

78.555 111.657 61.202 60.030 4 6

82.087 122.247 86.906 86.744 4 6

106.279 164.114 117.585 118.177 3 5

87.335 129.772 139.424 140.618 4 6

BM: Hong et al. (2016) MAE RMSE AIC BIC Peak Saturation

375.401 451.235 45.968 43.264 8 –

BM with integration constant: Kim and Hong (2015) MAE RMSE AIC BIC Peak Saturation

221.274 328.577 46.064 42.459 4 7

81.246 113.443 63.360 61.798 4 6

71.365 108.752 87.268 87.052 4 5

70.999 108.735 112.175 112.964 4 –

72.913 109.088 137.604 139.196 4 5

GGM: Guseo et al. (2017) and Guseo and Guidolin (2009) MAE RMSE AIC BIC Peak Saturation

228.514 396.273 49.188 44.682 5 6

76.293 95.446 63.633 61.680 4 5

53.482 68.707 82.839 82.569 4 5

53.560 68.737 105.920 106.906 4 5

53.200 68.493 129.365 131.354 4 5

Cumulative sales GBM: Bayesian estimation with informative prior MAE RMSE AIC BIC

652.693 719.268 52.765 48.258

471.828 571.312 81.527 79.574

43.324 48.964 78.097 77.826

41.226 54.241 101.657 102.643

42.550 53.337 123.863 125.852

BM: Bayesian estimation with informative prior MAE RMSE AIC BIC

629.017 688.621 48.504 45.800

2334.695 2922.685 93.850 92.678

741.023 795.997 113.136 112.973

593.206 642.547 142.153 142.745

466.663 511.392 169.594 170.788

81.989 96.008 59.692 58.520

74.699 95.014 83.378 83.215

164.593 232.727 123.873 124.464

87.73 109.318 135.651 136.844

BM: Hong et al. (2016) MAE RMSE AIC BIC

953.132 1212.700 51.899 49.195

BM with integration constant: Kim and Hong (2015) MAE RMSE AIC BIC

984.647 1283.300 54.239 50.633

115.027 127.467 64.526 62.964

59.001 77.195 82.470 82.254

59.619 78.266 106.257 107.046

57.719 75.331 129.458 131.050

(continued on next page)

strategy. On the other hand, if CNG is an inelastic product, a 5% price decrease will have a negative effect on the income from consumption. The manager needs to analyze the total effect on the project, which comprises the acceleration in adoption rate versus the possible reduction in income from consumption. 5. Concluding remarks This study has introduced a particular Bayesian estimation process for the generalized Bass model for sales forecasting. Using several simulation exercises, we found that this process generates rapidly-adapting forecasts with desirable properties. These forecasts predict the total and one-step-ahead sales information for a single product accurately. As our three applications to different product types have shown, information on similar products or homogeneous markets can be used to infer future adoption values even when no historical data on the product of interest are available (pre-launch forecast), as long as there are sufficient data on the similar products. The implications for managers and decision makers are then substantial. Marketing planners could collect data on sales and other variables for products similar to the one of direct interest and have a systematic way of combining this information in order to obtain a reliable forecast. In addition, the expert survey-based method proposed by Kim et al. (2013) may help in eliciting hyperparameters. These two pieces of information (reference markets and experts) can then be used to get the final hyperparameter sets. However, we should say that once the sales peak time has been reached, the BM and GBM are overcome by more flexible alternatives such as GGM or BMIC. We estimate these more flexible specifications using traditional approaches, but future research should consider Bayesian estimation. This research is also of methodological interest, since it explores the adaptation of posterior distributions to data availability and the effect of a Scaled-Beta2 prior distribution for the scale parameter, which has not been used widely in the literature but shows good properties. Finally, we have to acknowledge that our Bayesian proposal requires programming abilities for its implementation, whereas the frequentist methodologies (NLS and ML) are available easily through friendly graphical user interfaces. Acknowledgment The research was partly supported by Universidad EAFIT (Convocatoria Proyectos Internos 2018) grant 828000044.

Please cite this article as: A. Ramírez-Hassan and S. Montoya-Blandón, Forecasting from others’ experience: Bayesian estimation of the generalized Bass model. International Journal of Forecasting (2019), https://doi.org/10.1016/j.ijforecast.2019.05.016.

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A. Ramírez-Hassan and S. Montoya-Blandón / International Journal of Forecasting xxx (xxxx) xxx

Fig. 13. Scenario analysis: compressed natural gas. Notes: Alternative scenario where the price decreases or increases by 5% relative to the actual price path. Bayesian estimation with informative priors is used to construct the chains and predictive intervals for each subsample up to t.

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Andrés Ramírez-Hassan holds a Bachelor’s degree in Economics and a Master’s degree in Economics from Universidad Nacional de Colombia, a Master’s degree in Finance from Universidad EAFIT and a Ph.D. in Statistical Science from Universidad Nacional de Colombia. He has held faculty positions at Universidad Nacional de Colombia and Universidad de los Andes, and is currently affiliated with Universidad EAFIT as a professor and researcher. His main research interest is in Bayesian econometrics with applications to topics such as welfare analysis, crime, marketing, and finance.

Santiago Montoya Blandón holds a Bachelor’s degree and a Master’s of science degree in Economics from Universidad EAFIT and is currently an Economics Ph.D. candidate at Emory University. His research interests are in econometric theory and Bayesian econometrics, along with microeconometric applications.

Please cite this article as: A. Ramírez-Hassan and S. Montoya-Blandón, Forecasting from others’ experience: Bayesian estimation of the generalized Bass model. International Journal of Forecasting (2019), https://doi.org/10.1016/j.ijforecast.2019.05.016.